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Keywords:

  • spatial variability;
  • water content;
  • polythermal glacier;
  • cold surface layer;
  • Storglaciären;
  • Sweden

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[1] The volume fraction of liquid water in temperate glacier ice is important not only for the flow of glaciers and the analysis and processing of ground penetrating radar data from glaciers but also for the stability of the thermal layering in polythermal glaciers. However, little is known about the spatial variations of water content in glaciers. We use relative backscatter strength of ground-penetrating radar signals to estimate the spatial distribution of water content close to the cold-temperate transition on Storglaciären, northern Sweden, in an area close to the equilibrium line. The values of relative backscatter strength are calibrated using determinations of absolute water content from temperature measurements across the cold-temperate transition and the thermodynamic boundary condition at the freezing front. The results show a water content of 0.80%, 0.75%, and 0.58% at three calibration points and a mean water content of 0.8% with a standard deviation of ±0.26% for the extrapolated water content. The extrapolated water content shows a distinct pattern, with lower water content on one side of the glacier center line and higher water content on the other side, with higher water content on the northern side. We hypothesize that the different water contents result from the fact that the ice on either side of the center line originates from different cirques, thus implying spatial variations in the entrapment of water in the firn-ice transition process in the different cirques.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[2] Information on the absolute water content in temperate and polythermal glaciers is needed for many purposes in glaciology. The water content is important for the flow of glaciers because the deformation rate of ice is strongly dependent on the water content [Duval, 1977; Lliboutry and Duval, 1985]. It is also important to accurately convert the travel time of ground-penetrating radar (GPR) signals into depth estimates because the velocity of radar waves is strongly dependent on the water content [e.g., Daniels, 1988]. Typically, a constant velocity is assumed in glaciological studies, but this assumption will be incorrect if water content changes spatially or with depth [Murray et al., 2000a]. The water content also has implications for the stability of the cold-temperate transition surface (CTS) in polythermal glaciers [Hutter et al., 1988; Blatter and Hutter, 1991; Moore et al., 1999; Pettersson et al., 2003]. To understand the spatial variation in thickness of the cold surface layer and any changes in the thickness of cold surface layer, we need to know the liquid water content at the CTS and its spatial variation [Pettersson et al., 2003].

[3] Liquid water in glaciers may originate from four possible sources [Paterson, 1971; Lliboutry, 1976]: (1) adjustment of the pressure melting point due to changes in overburden pressure heats or cools temperate ice, where the energy involved is from freezing or melting of liquid water within the ice; (2) melting from strain heating; (3) water entering the glacier at the surface; and (4) water trapped in the ice as water-filled pores in the firn close off at the firn-ice transition in the accumulation area.

[4] It is likely that some or all of these sources vary spatially over a glacier, causing spatial variations in the water content. However, their relative importance for the total water content is not completely clear. Paterson [1971] argued that most of the liquid water in glacier ice originates from the entrapment of water at the firn-ice transition since strain heating would only contribute to the overall water content close to the margins where strain rates are high, while the other sources were considered small. In contrast, Hutter et al. [1988] used thermomechanical modeling to demonstrate a strong dependence of water content on strain heating, even at relatively shallow depths.

[5] The methods used to estimate the water content in ice could be divided into thermodynamic and remote sensing methods. The thermodynamic methods involve the use of an adiabatic calorimeter on retrieved ice cores [Dupuy, 1970; Vallon et al., 1976; Duval, 1977] or in situ calorimetric measurements, where the water content is determined from the propagation speed of an artificial freezing front in the ice and from the boundary condition at the freezing front [Hutter et al., 1990; Zryd, 1991]. The remote sensing techniques often use radar wave velocity analysis [Macheret et al., 1993; Moore et al., 1999; Murray et al., 2000a] or the backscattered power in radargrams [Bamber, 1988; Hamran et al., 1996; Macheret and Glazovsky, 2000] to estimate water content.

[6] Volumetric water content in glacier ice has been estimated by many different methods to be in the range from 0 to 9%, and values are summarized in Table 1. Very few of these investigations have emphasized spatial variations. Benjumea et al. [2003] found a lateral variation in water content in one seismic sounding profile, with the maximum water content in areas of largest ice thickness. Macheret and Glazovsky [2000] deduced both a seasonal variation in water content of 2.3% and a spatial variation of 1.7–11.9% from power reflection coefficients of GPR signals. However, it is problematic to model the absolute scattering power from the water voids in the ice because it is necessary to know the size and shape of the water inclusions and actual coupling between antenna and ground [Bamber, 1988; Hamran et al., 1996]. Instead, Hamran et al. [1996] tried to circumvent these problems by using the backscattered power relative to a reference point to estimate water content distribution in a polythermal glacier; however, they did not have an absolute calibration value at the reference point.

Table 1. A Compilation of Results From Previous Determinations of Water Content in Temperate and Polythermal Glaciers Using Different Techniques
W, %equation image,a %Depth, mMethodLocationReference
  • a

    Mean water content.

0.4–3.22.228–162GPR and seismic analysisJohnson's GlacierBenjumea et al. [2003]
1.0–1.5 799thermodynamic analysisJakobshavn isbrLüthi et al. [2002]
0.2–0.3 0–28GPR wave velocity analysisFalljökullMurray et al. [2000a]
2.4–3.3 28–95   
0.1–0.2 95–112   
2.8–4.1 ∼100GPR wave velocity analysisBakaninbreenMurray et al. [2000b]
2.8–9.1 variousGPR reflection coefficients21 Svalbard glaciersMacheret and Glazovsky [2000]
0.5–7.62.05–90GPR wave velocity analysisHansbreenMoore et al. [1999]
0.7–1.5 basal icein situ calorimetricEngabreenCohen [1999]
0.1–3.41.7127–368GPR wave velocity analysisHansbreen, Fridtjovbreen, and Abramov GlacierMacheret et al. [1993]
0.0–1.10.2152–247 Hansbreen, Fridtjovbreen, and Abramov Glacier 
0.5–3.41.70–220 Hansbreen, Fridtjovbreen, and Abramov Glacier 
0.5–1.5 basal icein situ calorimetricFindelengletcherZryd [1991]
0.0–3.00.70–220calorimeterGlacier d'ArgentièréLliboutry and Duval [1985]
0.2–1.3 30–187calorimeterVallée BlanceVallon et al. [1976]
0.2–0.9 60vein volumetric calculationsBlue GlacierRaymond and Harrison [1975]
0.4–0.7 22–55calorimeterGlacier de Saint-SorlinDupuy [1970]
<0.7 34–54calorimeterVallée BlanceJoubert [1963]

[7] In our study we determine the absolute water content with good accuracy and investigate the spatial variability of the water content over Storglaciären, northern Sweden, and strive to explain causes of this spatial distribution. To do so, we use the Hamran et al. [1996] method to calculate water content distribution relative to a reference point in 18 closely spaced GPR profiles across the glacier (Figure 1). To establish the absolute water content, we use an in situ calorimetric method to determine the water content at the reference points, but instead of introducing an artificial freezing front, we use the rate of migration of the CTS and determine the water content from the thermodynamic relations in the cold surface layer. We estimate the CTS migration rate from the displacement of the freezing front along thermistor strings installed across the cold-temperate transition. A similar technique was used by Lüthi et al. [2002] to establish water content at the CTS, but they did not measure the migration rate directly but instead relied on ice flow modeling for determining the migration rate. To be able to interpret the spatial pattern in water content, we also measure the ice velocity in a grid of 46 stakes over the upper part of the ablation area as ice flow influences the amount of water produced by strain heating and hydrostatic pressure changes.

image

Figure 1. Location map for the GPR profiles, the three thermistor strings (CTS1, CTS2, and CTS3), and ice velocity stakes. The GPR profiles are labeled 1–18.

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[8] We show that there is a distinct spatial distribution of water content beneath the cold surface layer on Storglaciären. The mean water content at three thermistor strings was 0.7%, while the extrapolated mean water content over the study area was 0.8% with a standard deviation of ±0.26%. We propose that this distribution is due to variation in water content in newly formed ice. This is because ice originating in different cirques may have a markedly different water content, implying spatial variations in the firn-ice transition process.

2. Methods

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

2.1. Water Content at Reference Points

[9] The one-dimensional transition condition at the CTS is

  • equation image

where dθcts/dz is the vertical temperature gradient just above the CTS in the cold surface layer side and Cp, L, and κ are the specific heat capacity, specific latent heat of fusion, and thermal diffusivity of ice, respectively. Wcts is the volume fraction of liquid water in the temperate ice arriving at the CTS that needs to freeze to let the CTS change position, and am is the volume flux of ice through the CTS. The volume flux is equivalent to the change in position of the CTS over a unit area (i.e., migration rate of the CTS) and is a combination of the upward velocity of the ice at the CTS caused by ablation at the surface and the rate at which ice refreezes at the base of the cold surface. Some heat conduction into the deeper temperate ice from the freezing front is possible because of the positive temperature gradient caused by the increasing pressure melting point with depth. In equation (1) we assume that the heat conduction from the freezing front into the temperate ice is limited. From equation (1) we could determine the water content at the CTS if we knew the temperature gradient in the cold surface layer and the migration rate of the phase boundary.

[10] We determined the temperature profile in the cold surface layer on Storglaciären by inserting thermistor strings and the migration rate of the CTS by recording changes in the depth of the freezing isotherm relative to the thermistors with time. The temperature profiles were measured at three locations (Figure 1) using Fenwal UNI-curve (192-502LET-A01) thermistors spaced 3 m apart to a depth of 30 m and 0.5 m apart between 30 m and 40 m. The resistances of the thermistors were measured and stored by a data logger every 2 hours using a three-wire half-bridge configuration. The thermistors have a standardized relationship between temperature and resistance, with a curve tolerance of ±0.5°C. The accuracy of the individual thermistors was improved by calibration in a mixed ice/water bath at 0°C, which resulted in an accuracy of ±0.05°C. The data logger has a resolution of ±67 μV, which corresponds to ±0.02°C in calculated temperature; hence the total accuracy of the temperature would be ±0.06°C.

[11] Ideally, we would record the time required for consecutive thermistors along the thermistor string to freeze into the cold surface layer. Knowing the time and the thermistor separation, we can estimate the migration rate of to the CTS relative to the thermistor string. However, the migration rate is presumably slow, roughly equal to the surface net balance, on the order of ∼1 m yr−1. This would require thermistors more closely spaced than ours to record the migration with accuracy. Furthermore, it is difficult to pinpoint when the ice surrounding the thermistors becomes cold. We have therefore chosen a different approach to determine the migration rate. We evaluate the thermodynamic equation for cold ice analytically and find the best fit to the solutions using the measured temperature profiles in the cold part of the cold surface layer. Extrapolating the fitted temperature profile in the cold surface layer to the melting point allows us to find the freezing isotherm position on the thermistor string. Comparing the results from the regression based on values for the temperature profiles at different times yields a change of the CTS position along the thermistor string over the considered time span.

[12] The one-dimensional thermodynamic equation in steady state for cold ice is

  • equation image

[Hutter et al., 1988], where w is the vertical velocity component. Assuming a constant vertical velocity, w(z) = w, performing a separation of variables and integrations of equation (2) yields

  • equation image

where G0 = dθcts/dz. The simplification that w(z) is constant through the cold layer is reasonable since the cold surface layer is about 35 m thick (∼15% of total ice thickness) at the thermistor string location and large changes in the vertical velocity component are not expected in the uppermost part of the ice column. The assumption with constant vertical advection at depth and steady state situation for the temperature profile is probably not correct for a real situation, but it simplifies the solution and is feasible for a first investigation, as in our case.

[13] The measured temperature yields θi(zi), i = 1…n, where n is the number of thermistors that are below the pressure melting point. Defining z = 0 at CTS and using z axis positive upward makes it possible to determine the position of the thermistors relative to the CTS by finding the best fit for θi(zi) and by using a Newton iteration for the equation

  • equation image

where A = κG0/w and B = w/κ. The regression yields A and B, so the temperature gradient at the CTS would be G0 = AB, and since θ(zcts) = 0, −C = A, A would be the displacement of the thermistors relative to the CTS. The vertical velocity, w = κB, is not necessarily the true vertical velocity at the site but is rather a tuning parameter. Repeating the regression for temporal mean values of the measured profile yields a change of the CTS position along the thermistor string and is a measure of the migration rate of CTS. We use mean values over a certain time span instead of sample values to reduce noise in the temperature data. The regression also gives the temperature gradient on the cold side of CTS and enables us to solve equation (1) for the water content at CTS.

2.2. Determining the Water Content From GPR Data

[14] The calculated water content at the thermistor strings provides reference points for calculating the water content from backscatter intensity in GPR profiles as described by Hamran et al. [1996]. On the basis of the well-known radar equation, Hamran et al. [1996] showed that the water content W relative to the reference point can be expressed as

  • equation image

where Pr, Rr, and Wr are the returned power, range, and water content, respectively, at a reference point (Wr is set to 1 if relative calculations are made). P and R are the returned power and range for an arbitrary point in the radargrams, and α is the attenuation in the media (neper m−1).

[15] We acquired 18 parallel GPR profiles with a separation of about 5 m in late April 2001 under dry snow conditions (Figure 1). The center profile passed over the CTS2 thermistor string, while profiles 4 and 16 passed the CTS1 and CTS3 thermistor strings, respectively (Figure 1). The GPR was mounted on a sled pulled by a snowmobile with an average travel speed of <5 km h−1, and the profiles were positioned by differential GPS measurements, storing positions for every 20th trace, corresponding to about 10–12 m of travel. Positions for GPR traces recorded between GPS positions were found by linear interpolation between the recorded positions. Our GPR system is a continuous wave stepped frequency radar using synthetic aperture radar processing and is described in detail by Hamran and Aarholt [1993] and Hamran et al. [1995]. The system is flexible and allows the user to determine the desired frequency and bandwidth for any specific task in the range of 0.03–6 GHz. We have used 700–900 MHz with a high-gain log-periodic antenna (AEL APN106AA) collecting 2 traces s−1. The collected frequency response along the profiles was transformed to the time domain using an inverse Fourier transform with no further processing of the data.

[16] We used the acquired GPR profiles to extrapolate the water content determined at the thermistor strings. We have chosen to use the water content determined at the center thermistor-string (CTS2) because the position of this thermistor string was located with highest accuracy (<1 m) along the GPR profile. The other two thermistor strings were located within 2 m along the profiles and were used as verification points for the interpolation.

[17] An important factor for estimating water content using equation (5) is the attenuation factor, which can be expressed as

  • equation image

where ω is the angular frequency and μ and ɛ are the magnetic permeability and permittivity of the media, respectively [Daniels, 1988]. tan δ is the so-called loss tangent, which is the attenuation resulting from dielectric and conductive losses that occur when an electromagnetic wave propagates through the ice. The loss tangent, and consequently the attenuation, depends on both frequency and temperature [e.g., Daniels, 1988]. Moreover, the attenuation in ice also depends on the presence of other dielectric media within the ice; water is the most important because of the large difference in dielectric properties between water and ice [Johari and Charette, 1975]. Impurities also affect the attenuation greatly because they affect conductive losses. However, glacier ice formed through typical temperate metamorphic processes [e.g., Schneider, 2000], and ice loses much of the deposited soluble impurities as they are washed out by percolating melt water; this results in very low conductivity of temperate ice, comparable to values for pure ice [Glen and Paren, 1975]. Chemical analysis of ice in the cold surface layer support this assumption (R. Pettersson, unpublished data, 1997). Thus we can assume that the effect of conductive losses on attenuation is small in temperate and polythermal glaciers with wet metamorphic firn-ice transformation. This implies that the attenuation would vary with depth in polythermal glaciers, depending only on temperature and the presence of free water.

[18] The attenuation down to the CTS is controlled primarily by the temperature in the cold ice. The coldest ice is found during late winter and spring in the uppermost part of the cold surface layer, where the ice temperature experiences large seasonal variations [Schytt, 1968]. Temperature measurements on Storglaciären show that the temperature in the uppermost part of the cold surface layer is not affected by seasonal variations is about −3°C [Schytt, 1968; Hooke et al., 1983; Pettersson et al., 2003]. Warren [1984, Table 1, pp. 1219–1224] has compiled and tabulated values for the dielectric absorption in pure ice at different temperatures over a wide frequency range. Warren's [1984] data indicate that an appropriate value for the attenuation at our GPR center frequency and at −1.5°C would be 0.043 dB m−1, with a maximum variation of ±0.008 dB m−1 due to temperature variations of ±1.5°C. This variation results in a deviation of calculated water content of ∼0.08% at the CTS (∼35 m depth). Hence as a first approximation, we use a constant attenuation value of 0.043 dBm−1 in the cold ice. This value is also consistent with determination of the attenuation on polythermal glaciers and cold glaciers reported in the literature [Robin et al., 1969; Smith, 1972; Bamber, 1987].

[19] The influence on the attenuation from the presence of small quantities of free water in temperate ice makes it difficult to estimate an accurate attenuation value for this ice without knowing the water content in the ice. This limits our use of equation (5) to estimate the water content only at the CTS or a limited depth below the CTS with reasonable certainty. Below the CTS the uncertainty increases rapidly due to possible variations in attenuation caused by unknown water content variations. This was not recognized by Hamran et al. [1996], who used the same attenuation for the whole ice thickness.

[20] The increase in uncertainty below the CTS can be estimated by assuming an upper limit of water content and by using the corresponding attenuation for the temperate ice in estimating the water content distribution. A lower limit is given by the attenuation value for dry ice at the melting point. The spread between these two limits indicates a maximum uncertainty in the calculated water content obtained from equation (5). The bulk dielectric absorption of temperate ice with a certain fraction of water can be modeled using a complex dielectric mixture and equation (6). Selecting from several mixing models [van Beek, 1967], we have used the complex Bruggeman-Hanai mixing formula for randomly dispersed spherical inclusions [Hanai, 1936], which has been widely used to describe moisture content in many geological media [e.g., Nguyen, 1999]. We have chosen this model because of its ability to handle complex permittivity. As the constituent parts in the mixture, we use the complex permittivity of pure ice given by Warren [1984] and the complex permittivity for pure water based on calculations by Ray [1972]. The calculation gives a difference in attenuation of ∼0.1 dB m−1 between a water content of 3% and <0.1%. This would cause an uncertainty of ∼10% of the estimated water content 2 m below the CTS.

2.3. Interpolation of Water Content

[21] The calculated water content in a 1 m zone below the CTS was averaged over depth in each radargram. The CTS was determined and digitized using the same technique as in Pettersson et al. [2003]. Each profile contains 800–1500 averaged values along the profile, and a subset of 100 randomly chosen points was extracted to reduce further processing efforts. Tests with different numbers of random points (e.g., 50, 200, and 300) did not significantly change the results.

[22] The subset of data was interpolated onto a spatially regular distribution with a grid size of 20 m using variograms adopted for undulating surfaces [Dagbert et al., 1984] and ordinary kriging interpolation [Deutsch and Journel, 1998]. We used a linear variogram model without nugget effect, thus restoring the original values at the raw-data locations. The data show strong anisotropy, and we used an anisotropy ratio of 2.5 for the slope of the variogram model.

2.4. Ice Velocity Measurements

[23] Ice velocity was calculated between surveys in May 2001 and May 2002 for 46 stakes in a 100 × 100 m grid (Figure 1). The stakes were surveyed with single frequency DGPS (Trimble 4600LS) and a baseline of ∼2 km, giving an accuracy of ±1 cm in the horizontal direction and ±5 cm in the vertical direction.

[24] Velocities at the stakes were interpolated onto a 20 × 20 m grid using a spline interpolation method [Sandwell, 1987]. In the interpolation procedure we set the glacier perimeter to zero velocity, which is reasonable since the glacier is frozen to its bed in a ∼100 m wide zone along the margins [Pettersson et al., 2003]. Horizontal strain rates were calculated from the derivatives of the interpolated surface velocities, while vertical strain rates were calculated from the horizontal divergence assuming incompressible ice. Effective strain rates were calculated neglecting the vertical shear strain rates (εxz, εyz), which are assumed to be small at the surface.

3. Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

3.1. Temperature Measurements and CTS Migration

[25] Figure 2 shows monthly mean values for each thermistor string. Tests with other averaging time windows (e.g., 2 weeks and 2 months) gave similar results. Sporadic high-frequency noise on some thermistors, caused by a malfunctioning multiplexer, was reduced using a Savitzky-Golay low-pass filter [Orfanidis, 1996] before averaging.

image

Figure 2. Monthly mean values for the three thermistor strings CTS1, CTS2, and CTS3. Shading indicates the time transgression from October 2001 (light gray) to October 2002 (dark gray). The dashed line in the plot is the pressure melting point.

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[26] The displacement of the CTS over time along the thermistor strings is shown in Figure 3. The migration rate was found by solving equation (4) to determine the depth at which the temperature profile intersected the melting point for each of the monthly mean temperature profiles in Figure 2, excluding the thermistors affected by seasonal variation (<15 m depth). The coefficient of determination, r2, for all the fits was >0.98. The gradient in Figure 3 is an estimate of the migration rate along the thermistor string. The initial large CTS displacement is due to freezing in the borehole and recovery of the temperature field around the thermistors. After an initial 3–4 months the gradient becomes roughly constant, giving us confidence in the determination of the migration rate. The average migration rate, temperature gradient on the cold side of the CTS, and water content at the three thermistor strings are given in Table 2. The error limits of the temperature gradients and migration rates is calculated from uncertainties in temperature measurements and from the 95% confidence bounds for the fitting coefficients in equation (4).

image

Figure 3. Migration rate of the CTS along the thermistor strings found from the intersection of the pressure melting point and the temperature profiles. The depth scale is defined from the bottom of the thermistor string to make the calculation independent of any surface melt. The gradient of the curve is a measure of the migration rate of the CTS with time. The dashed, solid, and dotted lines are CTS1, CTS2, and CTS3, respectively.

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Table 2. Migration Rate, Temperature Gradient, and Water Content at the Three Thermistor Strings
 CTS1CTS2CTS3
Migration rate, m yr−1−1.02 ± 0.2−1.04 ± 0.3−0.95 ± 0.3
Temperature gradient, °C m−1−0.032 ± 0.01−0.035 ± 0.01−0.024 ± 0.01
Water content, %0.75 ± 0.060.79 ± 0.070.58 ± 0.08

[27] Figure 4 shows an example of GPR data processed with equation (5). The profile passes the CTS2 thermistor, and its location is indicated in the radargrams. Figure 4b is processed using an attenuation of 0.043 dB m−1 above the CTS and with an attenuation corresponding to a water content of 1% below the CTS. The uncertainty in the water content extracted from the processed radargram is ±0.25% at 2 m below the CTS due to uncertainties in the attenuation model and the water content at the reference point. Even if this uncertainty for the extrapolated water content is rather large, the extrapolation provides a good picture of the relative distribution of water content over the study area. Voids within the cold ice close to the CTS also become evident as bright spots in the figure. The apparent high water content at the uppermost part of the profile is due to the strong intensity associated with the air and ground waves at the surface.

image

Figure 4. Radargram for the profile passing over the CTS2 thermistor string. (a) Unprocessed radargram, with the thermistor string marked with a vertical line. The CTS is marked with a dotted line. (b) Extrapolated water content using equation (5) and the water content at CTS2 as the reference value.

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[28] The calculated water content values along the CTS in all profiles are interpolated into a spatial distribution given in Figure 5, with CTS2 as reference point. We also processed the profiles using arbitrary reference points (with unknown water content) along the CTS to test the importance of choice of reference point. Even though we do not obtain an absolute scale using a arbitrary reference point, the relative distribution pattern remained regardless of which reference point was used, indicating consistency in the distribution.

image

Figure 5. Interpolated water content at CTS using kriging interpolation. Dashed contour lines (20 m interval) indicate ice thickness and solid contour lines (4 m interval) show CTS depth interpreted from radar profiles.

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[29] Figure 5 indicates that the water content is lower on the southern side of the glacier center line compared with the northern side, except close to the southern margin, where the water content again becomes high. This is also shown in Figure 6, where we have plotted the water content along all profiles as a function of distance across the glacier. The mean water content for all profiles is 0.8% with a standard deviation of 0.27%. The interpolated values close to the two thermistor strings, CTS1 and CTS3, that were not used as reference points for the calculations were 0.75 ± 0.2% and 0.58 ± 0.2%, respectively. Switching the reference point to any of CTS1 and CTS3 also reproduces the values at the other two thermistor strings. This agreement between the interpolated water content and calculated water content at the thermistor strings (Table 2) gives some confidence in the extrapolation of water content values using equation (5).

image

Figure 6. Water content for all profiles plotted as a function of distance (northing coordinates), transversing the glacier. Zero is set to the southern glacier perimeter. A sketch at the top of the figure shows the glacier surface, CTS, and bed in 1:1 scale. The dashed line indicates the position of the thermistor strings.

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3.2. Ice Velocity

[30] Contour maps of total horizontal ice velocity, horizontal strain rates, and effective strain rates are shown in Figure 7. Velocities are comparable with those found by Hooke et al. [1989], and the maximum speed is found just north of the center line. Our assumption of zero velocity at the glacier margins is supported by the negligible velocity at the northernmost stakes, while the southernmost stakes, located farther from the glacier margin, have velocities of ∼6 m yr−1. Effective strain rates are rather uniform over the area, except along the margins, where increased strain rates are due to friction against the valley sides.

image

Figure 7. Ice flow and the derivatives from repeated surveys of 46 stakes in a 100×100 m grid over a 1 year period (May 2001–2002). Values in the outer shaded area are extrapolated values, yet these values are reasonable since it is expected that the ice velocity decreases toward the glacier perimeter as assumed in the velocity calculations. (a) Total horizontal ice velocity. (b) Longitudinal strain rate at the surface. (c) Effective strain rates at the surface.

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4. Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[31] Our estimate of the water content, ∼0.8%, differs from mean values found in many temperate and polythermal glaciers using GPR techniques (Table 1). However, thermodynamic determinations of water content [Vallon et al., 1976; Duval, 1977; Lüthi et al., 2002] agree well with our values. The different results between remote sensing and thermodynamic methods may be explained by the existence of larger water-filled voids in the ice that are not sampled using thermodynamic methods but contribute to the determination using GPR techniques since the GPR method is often averaged over large depths [Murray et al., 2000a] and integrated over a volume illuminated by the GPR. The difference may also be caused by the dependence of GPR signal velocity on other factors such as ice density, impurities, and other scattering media besides water in the ice.

[32] Some water will not be frozen when the ice is advected into the cold surface layer. At grain boundaries and at three-grain intersections, liquid water may remain because of impurities, lowering the freezing temperature [Harrison, 1972]. However, the radar seems to pick up inclusions at a very distinct depth, which corresponds nicely with the CTS, except for some isolated larger voids of water [Pettersson et al., 2003]. This does not mean that ice above the inferred CTS is completely dry, just that the inclusions are substantially smaller in size and cannot be detected with the GPR.

[33] The observed spatial pattern in water content suggests local changes across the glacier in one or a combination of the four possible sources for liquid water in temperate glacier ice (changes in the pressure melting point, strain heating, water input at the ice surface, entrapment of water in the accumulation area [Paterson, 1971]). However, the water content in an element of ice arriving at the CTS is the cumulative sum of all possible sources along its flow path, from the accumulation area to the surface in the ablation area [Lliboutry, 1976]. To formulate the cumulative sum along a flow line, we can look at the energy balance of temperate ice.

4.1. Energy Balance of Temperate Ice

[34] We can treat temperate ice as a heat-conducting binary mixture of water and ice. The continuity equation for energy with the constitutive relations for internal energy equation image = Cpequation image + Lequation image and the nonadvective energy flux q = −k∇θ − νLW then yields

  • equation image

where D is the strain rate tensor, T′ is the deviatoric stress tensor, and tr denotes the mathematical operator trace. Variable ρ is the density of ice; ν is the moisture diffusivity, which is unknown. Equation (7) is not general since the extremely small contribution of kinetic energy is neglected, diffusion of heat is described as Fourier-type diffusion, which is realistic, and moisture diffusion is described as a Fickian-type diffusion, which is questionable. The temperature, θ, of temperate ice is assumed to be at the local pressure melting point of pure ice, and heat generated by mechanical or nonmechanical processes is immediately consumed by melting ice. Hydrostatic pressure, P, and the corresponding melting temperature can be written as a function of depth below ice surface, S. The hydrostatic pressure becomes P = Pr + ρg(Sz), and thus θ = θr − ϕ(PPr) = θr − ϕρg(Sz), where ϕ = 1.3 × 10−7 K Pa−1 is the pressure melting point depression, g is the acceleration of gravity, Pr and θr are a reference pressure and the corresponding reference melting temperature, respectively, and z is the vertical (parallel to gravity) coordinate. To rewrite equation (7) in the plane strain approximation (two-dimensional, following flow lines) in Cartesian coordinates (x, z), the corresponding temporal and spatial derivatives of temperature are substituted in equation (7):

  • equation image

where τxz and σxx are the shear stress and the longitudinal component of the deviatoric stress tensor, respectively, and u and w are the velocity components in the x and z direction, respectively. Note that the expressions for the thermal conduction and heat advection are not written in terms of temperature but in terms of surface geometry. The temperature is no longer an unknown field variable, and the corresponding expressions in equation (8) correspond to source terms for water content. With this, equation (8) is recovered as an advection-production equation for one unknown field variable, the water content W. To further simplify equation (8), heat and water diffusion are omitted [Blatter and Hutter, 1991], and steady state is assumed for the glacier geometry, moisture, and velocity fields:

  • equation image

Equation (9) is equivalent to

  • equation image

where WA and WB are the moisture contents at the starting and end points of the considered trajectory and integration is along the particle path. Q is heat produced by internal deformation. In our case the starting point A lies in the accumulation zone at a depth where water content is no longer affected by percolating melt water and the annual freezing front, and B is the position where the particle reaches the CTS. The initial value WA corresponds to the entrapment of water in the accumulation zone. The first and second term in the middle of equation (9) describe the water production by hydrostatic pressure changes and by strain melting, respectively. Hence by looking at the individual terms in equation (9), it is possible to get an estimate for the contribution of water content from strain heating and hydrostatic pressure changes. The contribution from entrapment of water at the ice-firn transition, WA, and the water input from the surface in the ablation area must be considered separately.

4.2. Changes in Hydrostatic Pressure

[35] To get a rough estimate of the hydrostatic pressure changes on the water content, we can simplify the first term in the middle of equation (9) by assuming a parallel slab model, hence ∂S/∂x = 0 and ∂W/∂x = 0. This reduces the equation to be dependent only on the pressure difference between the starting point A and end point B in equation (10), i.e., the difference in depth below the surface for points A and B. This is reasonable as changes in the hydrostatic pressure are opposite in the ablation area compared to the accumulation area. In the accumulation area the hydrostatic pressure increases as the ice is gradually buried, and the corresponding depression of melting point allows for the release of heat that becomes available for melting ice. In the ablation area the process is reversed. The ice is transported toward the surface, the overburden pressure decreases, the melting point is increased, and the ice has to be heated to remain temperate. The necessary heat is supplied by freezing of water, and thus the change in water content along the flow path is dependent only on the difference in hydrostatic pressure between the end points.

[36] Inserting appropriate values for ice density, gravity, specific heat, and latent heat of fusion in equation (10) yields a water content contribution of ∼0.7 × 10−3% m−1 of depth difference between points A and B. The difference in depth between the firn-ice transition in the accumulation area and the CTS is maximum ∼50 m based on comparison of measurements of the ice-firn transition depth [Schneider, 2001] and the cold surface layer [Pettersson et al., 2003]. This gives a water content of 0.035% from hydrostatic pressure differences. Although this is a rough estimate, it provides a good indication that hydrostatic pressure changes do not contribute significantly to the water content in the ice arriving at the CTS in the study area.

4.3. Strain Heating

[37] The second term in equation (10) is the contribution to the water content per unit time along a flow line by strain heating; assuming n = 3, it can be rewritten as

  • equation image

where A is a flow parameter and F is the creep response function of the effective stress; in our case, F = σxx2+ τxz2. Thus we can estimate the contribution from strain heating to the bulk water content along a flow line if we can approximate σxx and τxz and the transport time along a flow line.

[38] We can estimate σxx by assuming that ice velocities at the surface are representative also at depth. This is reasonable as the study area is close to the equilibrium line, so ice arriving at the CTS is transported only through the upper part of the ice column, where velocity does not change considerably with depth [Hanson, 1995]. This allows us to estimate σxx from the estimated ice flow derivatives at the surface. At the surface, τxz = 0, and thus the inverted Glen's flow law for plane strain becomes

  • equation image

where n = 3. For the flow parameter, A, we can use A = 28 × 10−16 s−1 kPa−3 found by Albrecht et al. [2000] for Storglaciären. Further, the shear stress can be approximated with the driving stress on the center line and near the surface assuming laminar flow; thus

  • equation image

where z is the depth for the calculations and ϕ is the surface slope [Nye, 1965]. An estimate of maximum depth for the flow lines ending in the study area can be taken from different flow models [Hanson, 1995; Pohjola, 1996], and mean surface slopes along the flow lines can be obtained from detailed maps of Storglaciären [Holmlund, 1996]. Although this is a very rough estimate of the water content contribution from strain heating, it is adequate for our purpose to estimate the order of magnitude of the contribution from strain heating. Moreover, the assumption of plane strain limits us to the central parts of the glacier since toward the margin the lateral shear stresses (τxy) become more dominant.

[39] The measured absolute values for longitudinal strain rates are between 0.005 and 0.02 yr−1 in the central parts of the study area (Figure 7b). We do not have any observations of strain rates in the accumulation area. However, Hooke et al. [1989] measured strain rates of ∼0.02 yr−1 in the lower part of the accumulation area, so the magnitude is comparable with strain rates below the equilibrium line. Inserting these strain rates, a mean surface slope of ∼7° and a maximum ice depth for the flow lines of ∼60 m [Hanson, 1995] into equations (11), (12), and (13) yield a contribution to the water content by strain heating of ∼0.5–1 × 10−3% yr−1 in the central parts of the glacier. However, these are maximum values since τxz decreases toward the surface.

[40] The time it takes for an ice particle to travel from the accumulation area to the study area can be estimated from the horizontal ice velocities and the average distance between the start and end points of the flow line. If we assume an average velocity of 17 m yr−1 (Figure 7a) and 800–1000 m of horizontal length for the flow line, estimated from twice the distance to the equilibrium line, we obtain a water content of maximum 0.1% from strain heating.

[41] This rough estimate illustrates that the contribution of strain heating to the overall water content in the central parts of the study area is small compared to the measured water content in the ice arriving at CTS. However, moving toward the margin of the glacier, we can expect the lateral shear stresses and strain rates to increase considerably due to drag along the valley walls. We also find increased strain rates (Figure 7c) and water content along the southern margin (Figures 5 and 6). The lack of increased water content along the northern margin is due to a local accumulation area that prevented us from extrapolating the GPR-derived water content into the marginal zone.

4.4. Water Input at the Surface

[42] Water entering the glacier through moulins and crevasses at the ice surface is concentrated into fairly well defined englacial conduits; percolation is considered negligible because the veins are small and may often be blocked by air bubbles [Lliboutry, 1971; Raymond and Harrison, 1975]. Furthermore, the cold surface layer on Storglaciären effectively inhibits any contribution from percolation from the surface in the ablation area.

[43] Larger voids in the ice, such as conduits, would create a strong reflection of the GPR signal and cause high values in the calculated water content. Such strong single reflections can be seen in Figure 4a and in the corresponding calculated water content in Figure 4b. These water content peaks may explain some of the variability seen in the interpolated water content map in Figure 5, but the reflections caused by voids are spatially restricted and cannot solely explain the distribution of the water content shown in the figure. Furthermore, the occurrence of voids within the cold surface layer may influence the calculation of water content at the CTS since some energy in the GPR signal is scattered in the inclusion and will not reach the CTS. It is possible that the englacial conduits are too small for detection with the GPR system (<0.2 m), but then a very high density (less than 0.2 m apart) of conduits would be needed to explain the large-scale variations in water content seen in Figure 5. Such a high concentration of conduits is not reasonable. It is unlikely that water input at the surface would substantially contribute to the bulk water content in the ice, other than, locally, in the form of conduits or large voids, especially on Storglaciären because of its polythermal structure.

4.5. Entrapment of Water in the Accumulation Area

[44] Paterson [1971] suggested that the most important source of liquid water in temperate glacier ice is entrapment in the transition from firn to ice in the accumulation area. Our results support this since none of the other origins of water content is plausible for explaining the observed water contents. Furthermore, both Vallon et al. [1976] and Dupuy [1970] found water content values below the firn-ice transition in the accumulation area of temperate glaciers that are comparable with our water content data from Storglaciären, thus also giving support for Paterson's conclusion.

[45] This implies that the spatial pattern of water content could be the result of large-scale variations in the entrapment of water in the accumulation area. Ice forms in two cirques on Storglaciären, which coalesce above the equilibrium line (Figure 1). An inferred boundary between the flow from the cirques is located just south of the center line in the ablation area [Hanson, 1995; Pohjola, 1996; Albrecht et al., 2000]. This suggests that the low water content in the southern part of the study area originates from the southern cirque, while the ice with higher water content comes from the northern cirque.

[46] The densification from wet firn to ice and the entrapment of liquid water is complicated and not fully understood, but the process depends largely on stress, capillary pressure in the firn, concentration of impurities, and initial firn grain size [Colbeck et al., 1978]. From our present data we cannot deduce the process responsible for the different water contents in the two cirques of Storglaciären. However, one hypothesis is that the cold wave during winter penetrates to or below the firn-ice transition in parts of the southern cirque that have lower firn thickness due to a lower accumulation rate, while the higher firn thickness in the northern cirque keeps the firn-ice transition temperate throughout the winter [Schneider, 2001]. The penetration of the cold winter wave to the firn-ice transition allows freezing of liquid water and formation of ice with a lower water content. Holmlund and Eriksson [1989] have observed by GPR surveys that the cold wave penetrates deeper than 25 m in the lower part of the southern cirque. As the accumulation rate increases toward the northern cirque, the penetration of the cold wave decreases, resulting in a continuous change in the entrapment of water. However, this is the present-day situation; we do not know if these conditions prevailed when the ice arriving at the CTS today was formed.

5. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[47] Our estimate of the water content in the temperate ice below the cold surface layer on Storglaciären agrees with earlier estimates of water content in temperate and polythermal glaciers using calorimetric methods. The water content at three locations determined from the transition condition at the CTS ranges from 0.6 to 0.8%. Using any of these reference points, we can extrapolate the water content at the CTS using relative backscattered power in GPR surveys. The extrapolation reproduces well the values of water content at the two locations not used as references, thus giving confidence in the extrapolation, even though the uncertainty in the extrapolated water content is rather high (±0.25%). This is due to uncertainties in modeling the spreading and attenuation losses of the GPR signal strength. The uncertainties increase in temperate ice because the attenuation is dependent on the unknown water content. This limits us to estimating the water content just below the CTS with some accuracy. Although our determination of water content is limited to the cold-temperate ice transition, it can provide a constraint to future modeling of the moisture content in the whole body of the glacier.

[48] The liquid water in the ice arriving at the CTS may have several origins, but it seems likely that the most important is entrapment of water as the firn is transformed into ice in the accumulation area, supporting Paterson [1971]. Changes of water content due to changes in pressure melting point or strain heating are found to be negligible, while percolating water from the surface is probably nonexistent.

[49] The spatial pattern of the extrapolated water content shows low and high water content areas on either side of the center line. We suggest that the difference in water content pattern is caused by ice originating in different cirques, with different amounts of entrapment of water at the firn-ice transition. The increased water content along the margins may be the result of the increased strain heating occurring close to the margins.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[50] We would like to thank Svein-Erik Hamran and Eldar Aarholt for stimulating discussions and for clarifying the processing of the GPR images in their work and Elsbeth Kuriger for ice velocity measurements. Regine Hock and Ann-Marie Berggren provided valuable comments on an early version of the manuscript. We also thank Luke Copland, Tavi Murray, and the Associate Editor Garry K. C. Clarke for careful reviews that helped to improve the manuscript. We also want to thank Roger LeB. Hooke for helpful suggestions. Any remaining ambiguities are the responsibility of the authors. R.P. acknowledges generous funding from the Göran Gustavssons Foundation. Funding was granted to P.J. from the former Swedish Natural Science Research Council (NFR).

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  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information
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Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information
FilenameFormatSizeDescription
jgrf53-sup-0001-tab01.txtplain text document1KTab-delimited Table 1.
jgrf53-sup-0002-tab02.txtplain text document0KTab-delimited Table 2.

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